Nonrational Modular Consistency and Spectral Densities

For a nonrational two-dimensional CFT, modular consistency is generally an equation for measures or distributions on a continuum of characters. A finite modular matrix is replaced by an integral kernel, and isolated states such as a vacuum must be separated from an absolutely continuous density. The free noncompact boson makes this replacement exact: its modular $S$ transformation is an ordinary Fourier transform.

Required background. Distributional correlators, zero modes, and normalization fixes the weak sense of equality and the volume conventions used below. Chiral blocks, sewing, and modularity supplies the rational finite-matrix construction being generalized.

Helpful background. Modular crossing and spectral bounds develops quantitative consequences once the spectrum, positivity assumptions, and kernel have been fixed.

Characters as a continuous basis

Let $\chi_\lambda(\tau)$ denote a family of chiral characters labelled by $\lambda$ on a contour $\mathcal C$ . A modular transform can take the form

\chi_\lambda(-1/\tau) =\int_{\mathcal C}d\nu(\lambda')\, S(\lambda,\lambda')\chi_{\lambda'}(\tau) +\sum_{a\in\mathcal D}S(\lambda,a)\chi_a(\tau).

The discrete set $\mathcal D$ is written separately because a delta function in a continuum density and an isolated representation can transform differently under analytic continuation. The formula is meaningful only after declaring:

the normalization of $\chi_\lambda$ ;
the measure $d\nu$ and contour $\mathcal C$ ;
the delta distribution associated with that measure;
any reflection identification, such as $\lambda\sim-\lambda$ ;
the test-function or function-space domain of the kernel;
the prescription for poles that cross during continuation.

A general torus amplitude may contain both discrete and continuous pieces:

\begin{aligned} Z(\tau,\bar\tau) ={}&\sum_{a,b\in\mathcal D} N_{ab}\chi_a(\tau)\overline{\chi_b(\tau)}\\ &+\int_{\mathcal C\times\bar{\mathcal C}} d\nu(\lambda)d\bar\nu(\bar\lambda)\, \rho(\lambda,\bar\lambda) \chi_\lambda(\tau) \overline{\chi_{\bar\lambda}(\tau)}. \end{aligned}

Modular invariance means equality after applying the kernels to this entire measure, including isolated terms and regulator-dependent volume factors. It need not be a pointwise equation for $\rho$ .

Exact Fourier transform for a noncompact boson

Use the convention

\chi_p(\tau)=\frac{q^{p^2/2}}{\eta(\tau)}, \qquad q=e^{2\pi i\tau}, \qquad p\in\mathbb R.

The Gaussian Fourier integral and $\eta(-1/\tau)=\sqrt{-i\tau}\,\eta(\tau)$ give

\begin{aligned} \int_{\mathbb R}dp'\, e^{-2\pi ipp'}\chi_{p'}(\tau) &=\frac{1}{\eta(\tau)} \int_{\mathbb R}dp'\, e^{\pi i\tau p'^2-2\pi ipp'}\\ &=\frac{e^{-\pi ip^2/\tau}} {\sqrt{-i\tau}\,\eta(\tau)}\\ &=\chi_p(-1/\tau). \end{aligned}

Thus the full-line modular kernel is

S(p,p')=e^{-2\pi ipp'}, \qquad d\nu(p')=dp'.

The state normalization remains $\langle p|p'\rangle=2\pi\delta(p-p')$ , so the partition-function density per unit target length is

\begin{aligned} \frac{Z(\tau,\bar\tau)}{L} &=\int_{\mathbb R}\frac{dp}{2\pi} |\chi_p(\tau)|^2\\ &=\int_{\mathbb R}\frac{dp}{2\pi} \frac{e^{-2\pi\tau_2p^2}}{|\eta(\tau)|^2}\\ &=\frac{1}{2\pi\sqrt{2\tau_2}\,|\eta(\tau)|^2}. \end{aligned}

This is modular invariant because $\tau_2\mapsto\tau_2/|\tau|^2$ and $|\eta(-1/\tau)|^2=|\tau|\,|\eta(\tau)|^2$ . Equivalently, Fourier orthogonality gives

\int_{\mathbb R}\frac{dp}{2\pi} e^{-2\pi ipp'}e^{2\pi ipp''} =\frac{1}{2\pi}\delta(p'-p''),

which reproduces the same $dp'/(2\pi)$ diagonal measure after transforming both chiral factors.

Because $\chi_p=\chi_{-p}$ , one may instead use $p\geq0$ . With bare half-line measure $dp'$ , the kernel becomes

S_+(p,p')=2\cos(2\pi pp').

Absorbing the factor of two into the measure or into normalized reflected characters changes this kernel. A comparison must therefore state whether the label space is $\mathbb R$ or $\mathbb R_+$ and how the endpoint $p=0$ is treated. The free-boson torus zero mode and modular factor are derived in Di Francesco, Mathieu, and Sénéchal 1997, §§10.1–10.2, pp. 337–343.

Four models, four kinds of data

The following comparison is a semantic summary of the chapter. It is designed to prevent a finite multiplicity, an extension class, and a continuum density from being entered into the same column as if they were the same kind of object.

Example	Central charge; spectrum and measure	Dilatation and pairing	Correlators and zero modes	Modular object and hypotheses	Evidence and decisive failure test
Yang–Lee minimal model $M(5,2)$	$c=-22/5$ ; two irreducible Virasoro families with discrete integer multiplicities; $h_{\min}=-1/5$ and $c_{\mathrm{eff}}=2/5$	$L_0$ is diagonalizable on irreducible modules, but the radial pairing cannot be positive because $\lVert L_{-1}\phi\rVert^2=2h_\phi\lVert\phi\rVert^2<0$	Ordinary power-law correlators; no continuum zero-mode measure; fusion $\phi\times\phi=\mathbf1+\phi$	A finite $2\times2$ character matrix and a finite rational fusion algebra; modular multiplicities are not descendant norms	Exact minimal-model representation data. Failure test: using $c_{\mathrm{eff}}$ in the Virasoro algebra gives the wrong null relations.
Rank-two logarithmic pair $(C,D)$	A discrete generalized eigenvalue $\Delta$ ; composition factors alone do not specify the nonsplit extension	$\mathcal D=\Delta\mathbf1+N$ with $N^2=0$ and $N\neq0$ ; $\langle CC\rangle=0$ , $\langle CD\rangle\neq0$ , so the pairing is degenerate on the eigenvector alone	$\langle DD\rangle$ contains $-2b\log(\mu\lvert x\rvert)\lvert x\rvert^{-2\Delta}$ ; no continuum is implied	Ordinary characters can miss $N$ and $b$ ; pseudo-traces or generalized torus amplitudes may be required in a specified logarithmic model	Exact Ward-identity result for the local pair, not by itself a complete modular CFT. Failure test: $D\mapsto D+\alpha C$ shifts the constant term but cannot remove $N$ .
Noncompact free boson	$c=1$ ; $p\in\mathbb R$ with $\langle p\lvert p'\rangle=2\pi\delta(p-p')$ and completeness $\int dp/(2\pi)$	$L_0$ is diagonalizable and the wave-packet pairing is positive; individual momentum states are not normalizable	Correlators contain $2\pi\delta(\sum_i p_i)$ ; the torus trace has target-volume factor $L$ , while $Z/L$ is finite	Fourier kernel $e^{-2\pi ipp'}$ on $\mathbb R$ , or a convention-dependent cosine kernel on $\mathbb R_+$ ; equality is distributional	Exact Gaussian transform. Failure test: omitting $dp/(2\pi)$ or dividing the charge delta pointwise produces a wrong volume factor.
Spacelike Liouville theory	$c=1+6Q^2$ with $Q=b+b^{-1}$ ; $P\geq0$ after reflection, with a declared delta normalization and DOZZ-weighted OPE integral	Principal-series $L_0$ is diagonalizable; reflection identifies $P$ and $-P$ ; positivity statements require the spacelike real- $b$ domain	No translation-invariant free zero mode; the interacting zero-mode integral is meromorphic. Four-point functions use a continuum contour plus residues after pole crossings.	Virasoro fusion and modular transforms are integral kernels whose measure depends on block/character normalization; degenerate limits become finite matrices	Exact special-function structure constants and kernel constructions under stated analyticity assumptions. Failure test: analytic continuation without crossed-pole residues disagrees with degenerate fusion and channel crossing.

The Yang–Lee entry follows Di Francesco, Mathieu, and Sénéchal 1997, §§7.3–7.4, pp. 211–221. The continuum and Liouville entries follow Ribault 2018, §§3.1 and 4.1.3, pp. 69–81 and 101–103, with the fusion-kernel hypotheses stated in Ponsot and Teschner 1999, §§1–3.

Positivity, the vacuum, and asymptotics

A continuum density can be nonnegative in a unitary theory, but its positivity is a statement about a measure:

\int_{\mathcal C}d\nu(\lambda)\, \rho(\lambda)|f(\lambda)|^2\geq0

for an appropriate test-function domain. It does not imply that a coordinate-dependent function called $\rho(\lambda)$ is invariant under reparametrization; $\rho$ changes with the Jacobian.

The vacuum also needs separate treatment. In a compact unitary CFT it is an isolated normalizable state. In a noncompact theory, the identity operator may exist while the corresponding constant target-space wavefunction is not normalizable in the direct-integral Hilbert space. A modular equation that inserts a vacuum delta function into a continuum must justify the measure and distributional domain rather than assume the compact formula.

High-temperature asymptotics depend on which lowest state appears in the regulated trace. For a nonunitary discrete theory this can replace $c$ by $c_{\mathrm{eff}}=c-24h_{\min}$ . For a continuum, a threshold density and its near-threshold behavior can multiply the leading exponential by powers of $\tau_2$ . Any asymptotic claim should therefore state the regulator, volume division, spectral threshold, and whether isolated states have been separated.

Modular consistency checks

Kernel normalization: compose $S$ with itself on a test function. For the full Fourier kernel, $S^2f(p)=f(-p)$ , the charge-conjugation action.
Measure: transform the complete $d\nu\,\rho$ , not just the character. Reparametrize the continuum and verify that the result is unchanged.
Vacuum and discrete terms: track isolated contributions separately and check whether they are normalizable states, distributional insertions, or residues.
Volume regulator: compare $Z_L/L$ at large target length rather than silently dropping $L$ .
Distributional equality: smear both sides with the same test functions before removing regulators.
Asymptotics: isolate the lowest discrete weight or continuum threshold and bound the remaining integral before applying a Cardy-like estimate.

The familiar finite Verlinde formula assumes a finite set of characters and an invertible finite modular matrix; its rational derivation is reviewed in Di Francesco, Mathieu, and Sénéchal 1997, §10.8, pp. 374–379. A continuous kernel may admit a generalized harmonic-analysis relation in a specified theory, but there is no universal instruction to replace matrix sums by bare integrals. Measures, distributional terms, and analytic continuations are theory-dependent.

For quantitative modular bounds and their additional positivity and gap assumptions, continue to Modular crossing and spectral bounds.

Exercises

Square the Fourier modular transform

Let

(Sf)(p)=\int_{\mathbb R}dp'\,e^{-2\pi ipp'}f(p').

Show that $S^2f(p)=f(-p)$ for a Schwartz function $f$ .

Solution

\begin{aligned} (S^2f)(p) &=\int dp'\,dp''\, e^{-2\pi ipp'}e^{-2\pi ip'p''}f(p'')\\ &=\int dp''\,\delta(p+p'')f(p'')\\ &=f(-p). \end{aligned}

The second modular transform is charge conjugation, as required.

Check the boson density under $S$

Verify directly that

\bigl[\sqrt{\tau_2}|\eta(\tau)|^2\bigr]^{-1}

is invariant under $\tau\mapsto-1/\tau$ .